Anyone can help me figure this Kelly problem out?

[blix177]

So Bankroll if $10k:
NBA Total for this game is, UNDER 200 +100,
OVER 200 1/2 +110.

Odds to tie at 200 for this game is 2% ( which represent losing one side out right.)

How much should I wager?
Both side has +EV.

Thanks in advance.

[blix177]

Bump =)

[Dunder]

Blix. This is not a Kelly issue, it is a very marginal Polish Middle i.e. you are laying (betting against) exactly 200 points.

You are essentially laying odds of +4198 (10,000 to win 244) on a +4900 outcome.

If you were to go ahead it would be 512.20 on the +100 and 487.80 on the +110 to have a get gain of 24.40.

[blix177]

I thought this would be a Kelly problem, because there are 3 possible outcome.
A. It goes under
B. It goes over
C. It hits 200 and I tie one bet and lose one bet. (odds it tie is around 2% of the time; base on the caculator.)

The spread between the two is generates an +EV. If I bet marginally all the time with these odds my bankroll should grow, but if i go all in each time, eventually I will break my bankroll because if it hits 200 just once I am down %50.

So question still remain how much should I be betting given a 10k bankroll.

[reno cool]

looks to me like you would be betting to win $37. 783 and 745
# are very rough, hopefully close


on 2nd thought, it might be twice that because you can only lose half.

[Justin7]

It is a Kelly problem. If you "dutch" it - 100 to win 100, 95 to win (roughly 105), you either
win 5 (98%)
or lose 95.
EP = 0.98 (5) - 0.02 (95) = 4.9 - 1.9 = 3
$3 profit with $195 capital: EV = 1.54

Odds of you winning are 98:2, or 49:1.

Your optimal risk amount would then be (49) (1.54) = 75.46% of your capital (split between the two sides). I'm doing this quickly, so pardon any mistakes... but this split would more than double your bankroll ever 50 opportunities.

Even a modest bankroll will max out the limits on the offering. Once your hedging does this, I'd figure out which side is the right side, and max that (hedging the other side to taste).

[blix177]

Thanks Justin!

But 75% of my bankroll would give me a heart attack. I think if I run into this I will do a 1/4 Kelly.

Also now i can play with the Kelly formula to figure out how to plug those number in.

[Dunder]

I thought this would be a Kelly problem, because there are 3 possible outcome.
A. It goes under
B. It goes over
C. It hits 200 and I tie one bet and lose one bet. (odds it tie is around 2% of the time; base on the caculator.)

The spread between the two is generates an +EV. If I bet marginally all the time with these odds my bankroll should grow, but if i go all in each time, eventually I will break my bankroll because if it hits 200 just once I am down %50.

So question still remain how much should I be betting given a 10k bankroll.

Based on the 0.44% edge you have, your total staked would be 16% of bankroll approximately.

i.e. USD 819.51 at +100
USD 780.49 at +110

to return 1639.02

This assumes the probability of 200pts is exactly 2%. If it is 2.1% the figures according to Kelly change dramatically. This is too marginal for Kelly, IMHO.

[blix177]

where:

• f* is the fraction of the current bankroll to wager;
• b is the net odds received on the wager (that is, odds are usually quoted as "b to 1")
• p is the probability of winning;
• q is the probability of losing, which is 1 − p.

So P=.98, Q=.02, B=1.05

{(.98)(1.05)-.02}/1.05 = 96% ????

Where did I go wrong?

[Dunder]

where:

• f* is the fraction of the current bankroll to wager;
• b is the net odds received on the wager (that is, odds are usually quoted as "b to 1")
• p is the probability of winning;
• q is the probability of losing, which is 1 − p.

So P=.98, Q=.02, B=1.05

{(.98)(1.05)-.02}/1.05 = 96% ????

Where did I go wrong?

b=1.0244

[Justin7]

I'm not sure the formula works quite right here. That is to determine the amount at risk. In this example, roughly half your stake is at risk of loss.

another way to solve this:
assume you risk x percent of your bankroll. Assume you win 49 times, lose 1 time. What x gives you the highest return?

I agree with Dunder's warning though - I think 2.0 is conservative for the push on an NBA total. 2.2% would be my best guess.

[Pancho sanza]

Whats the current market price? You need to know in order to figure it out.

Ill assume its 200 flat

Over 200.5 +110 $5800
Under 200 +100 $6090

You need to borrow $1890 here

Most you can lose is $5800, total potential win = 290

[taravat76]

I'm not a big believer in Kelly betting per se.

[u21c3f6]

Pancho has it correct. Full Kelly risk amount is 58% of bankroll at risk. However as Pancho points out, this hedge would require wagering more than your bankroll. The amount to wager with a 10K bankroll would be $4,878 on +110 and $5,122 on +100 for a 244 gain for a risk of $4,878. Of course this and any other amount combinations not exceeding your bankroll will always be a fractional Kelly.

The point about Kelly is to have a way to determine whether or not you are overbetting or underbetting. I also use it to determine if I am better off with smaller edge with a greater win % or vice versa. One does not necessarily have to bet strict Kelly to benefit from knowledge of Kelly.

Joe.

[Sinister Cat]

I just attempted this is in Excel, maximizing expected utility at full kelly, limited by bankroll size, using Solver, assuming probabilities of .505947 for <=199, .02 for 200, and .474053 for >= 201 (these seem to be the probabilities implied by the odds assuming vig of -110/-110, but I might be wrong). I got that the full bankroll should be wagered, $5259.47 on the under 200, $4740.53 on the over 200.5.

The proportions change if the total amount that can be wagered is limited to less than full bankroll: For example, if wagers are limited to $2000 total, then $1118.13 should be wagered on the under, $881.87 on the over.

It's very possible that I made a mistake in these calculations. I attempted to follow the methodology given by Ganchrow in this post:

http://forum.sbrforum.com/handicapper-think-tank/24713-using-kelly-determine-optimal-hedging-strategy.html

[GELATINOUS CUBE]

wouldn't have time to risk $100 to win $5 J7.

So did you win you're bet? Oh yeah you guys aren't really 'Players'

but it be like 2% push (win A or amount bet x zero or 0) and 49% under 200 win A x 2 (.49A2 + .2A)
and 49% win at the over 200.5 or (.49A2.1).

so solve for A, under bet would pay a return of, or an expected value of.... 100% (1A) and the over bet 102.9% (1.29A)

so the over bet would have better value at your win percent 49%, cuz it's +110 (pretty good money line price).

[GELATINOUS CUBE]

no not 1.29A actually 1.029A.

again slightly better EV on the over bet.

There's a nice difference in value in betting a +110 than a +100 especially if you are betting a lot of cash.

Yeah, it's raining so I'm in the tank

[GELATINOUS CUBE]

but yeah with a 10K bankroll, you are saying aim at $37 profit per bet, by playing both sides. Get kicked out of the casino, then what? Pick a side. You'd get robbed betting that much every day, but wishful thinking. Online, I guess it's possible.

You have a good book, if they are giving you +100/+110 where is this?? Matchy?

[GELATINOUS CUBE]

how about just betting the over at +110.

1.029 returns... so 2.90 on a 100 bet
vs. 37 on 7500 bet. or 4 on 750 or 800, ... 1 on 200 or 50 cents on 100 bet!
I'm off a little, but you'd be better betting that +110 every time

[u21c3f6]

I just attempted this is in Excel, maximizing expected utility at full kelly, limited by bankroll size, using Solver, assuming probabilities of .505947 for <=199, .02 for 200, and .474053 for >= 201 (these seem to be the probabilities implied by the odds assuming vig of -110/-110, but I might be wrong). I got that the full bankroll should be wagered, $5259.47 on the under 200, $4740.53 on the over 200.5.

The proportions change if the total amount that can be wagered is limited to less than full bankroll: For example, if wagers are limited to $2000 total, then $1118.13 should be wagered on the under, $881.87 on the over.

It's very possible that I made a mistake in these calculations. I attempted to follow the methodology given by Ganchrow in this post:
http://forum.sbrforum.com/handicappe...-strategy.html

Sinister Cat, your solution is different because you assigned different probabilities to the outcomes. While probably more mathematically pure, when I hedgeI don't look at it like that. I prefer to set my hedge to give me the same profit regardless of the results. That may not be "optimal" but it fits my personality. In your scenario assuming your probabilities, there are three outcomes, win 50.6% @ $518.94, lose 47.4% @ $44.89 and lose 2% @ $4,740.53 which results in an EV of: $146.47. This is higher than my scenario of 98% win @ $244 and 2% loss @ 4,878 which results in an EV of: $141.56. I am willing to give up the small difference in EV to "win" more consistently.

Joe.

[u21c3f6]

how about just betting the over at +110.

1.029 returns... so 2.90 on a 100 bet
vs. 37 on 7500 bet. or 4 on 750 or 800, ... 1 on 200 or 50 cents on 100 bet!
I'm off a little, but you'd be better betting that +110 every time

Good question! Why not just bet the +110? Here's why I don't.

This scenario will also help me answer previous posters that say why hedge, just bet the +EV side. First, when I hedge I don't necessarily know which "side" is the +EV side even though I may "know" that the hedge is +EV overall.

Assuming the scenario of winning +110 49% of the time, a full Kelly % would be 2.635% of bankroll. With a $10,000 bankroll, that would equate to a wager of $263.50. Win 49% @ $289.85 and lose 51% @ $263.50 for an EV of: $7.64.

The hedge I suggested ($4,878 on +110 and $5,122 on +100 for a 244 gain for a risk of $4,878) is not even full Kelly because full Kelly would require you to wager more than your bankroll. In addition, the hedge I am suggesting has you wagering on a -EV scenario (49% win @ +100) but the overall hedge results in an EV of: $141.56 or over 18 times greater than just wagering on the +EV side.

The greater the win % (the overall hedge must still be +EV), the greater the amount of your bankroll you can risk "safely". Therefore, even if your overall edge is reduced by using a hedge instead of just wagering on the +EV side, the greater amount that you can wager "safely", results in greater profits.

Joe.

[reno cool]

lets see what I'm missing here.
use $100 for simplicity. If you were to bet 50 on both sides your avg win would be $2.5
If you're to do this 50 times winning 49 you net 122.50 and lose 50 once. An expected return of 72.50 for 5000 wagered. Since you can only ever lose one side of the bet you're only really risking half that so 72.50 of 2500.... I guess your edge really is close to 3%. I was way off before.

[DoubleEM]

I'm going to take a stab at this for practice & critique using the OP's numbers. Please let me know if I made a mistake.

**************************************** *******

First Question: Should I bet so that I profit an equal amount regardless of whether the Over or Under wins? Or should I put the full profit on either the Over or the Under? Let's see:

Same profit on each side:

$97.56 bet on over yields $107.32
$102.44 bet on under yields $102.44

So when there is no tie, I'm guaranteed a return of $4.88/$200.00 = 2.44%.


Put all profits on the Over:

$100.00 bet on over yields $110.00
$100.00 bet on under yields $100.00

So when there is no tie, half the time I earn a return of $10.00/$200.00 = 5.00%, half the time I earn a return of $0.00/$200.00. So, on average I earn a return of 2.50%.


Now let's put all profits on the Under:

$95.24 bet on over yields $104.76
$104.76 bet on under yields $104.76

So when there is no tie, half the time I earn a return of $9.52/$200.00 = 4.76%, half the time I earn a return of $0.00/$200.00. So, on average I earn a return of 2.38%.


Conclusion: Put all of the profit on the Over.

**************************************** *******

Second Question: How much of my bankroll should I bet to maximize future growth?

Now, for simplicity, I am assuming that the total at 200 is efficient, so:

Over 200 occurs 49% of the time. It will profit 10.00% on my total amount risked.
Tie occurs 2% of the time. It will lose 100% of my total amount risked.
Under 200 occurs 49% of the time. It will profit 0.00% on my total amount risked, for no gain or loss.


This is the Kelly Equation:
Bankroll % to bet = (O*w-(1-w))/O
O = the odds received
w = probability of winning

I am going to ignore the Under at this point because it doesn't affect my bankroll if it wins. Now I just need to compare how often the Over occurs compared to a Tie in order to calculate w, the probability of winning.

w = probability of winning = Over occurs/(Over or Tie occurs) = 49.00%/(49.00% + 2.00%) = 96.08%

O = the odds received = amount won/amount risked = $10.00/$100.00 = .1

Bankroll % to bet = (O*w-(1-w))/O = (.1*.9608-(1-.9608))/.1 = 56.86%

**************************************** *******

So, the OP should risk $5686. Any bet on the Under is never at risk, so bet $5686 on the Over & $5686 on the Under. However, the OP only has $10,000, which means put $5000 on the Over & $5000 on the Under. OP will make $500 for every Over & lose $5000 if the total lands on 200. He will, on average, win $500 49 times for every 2 $5000 losses, making this a very profitable play in the long run (assuming his %'s on the Over, Under, & Tie are correct).

Anyone agree or disagree?

[AaronTheGambler]

DoubleEM, I think your first answer might be in err because your assuming the odds to either events are 50/50. I am assuming the chance that Over would be less than 50% hence the +110 juice.

[DoubleEM]

I considered that, but took the easy way out. Maybe I'll calculate a fair line tomorrow and see what the true percentages should be and try part 1 again. It will still be profitable, though.

[DoubleEM]

This will affect part 2 also because the Over will win less times compared to the Tie. But, again, I'm thinking it will still be profitable. I'll try it tomorrow. Thanks Aaron

[beerman2619]

anyone like beer